Also - don't ignore car hire. Rhodes last year was my first attempt at driving on the wrong side of the road - and I had no problems. The road surfaces aren't bad. (remember the sharp intake of breath when they quote you a price - you should find it will reduce!!)
Rhodes isn't just warm in June, if last year is anything to go by (40 degrees daytime temps still late 20s early 30s in the evenings). If we didn't have the airconditioniong in both the room and the car we may have struggled.
You will probably be told you should've booked for the week earlier - we'll be there for the open air rock festival on the weekend before you arrive.
My preferences are for areas like Lindos, Stegna, Haraki etc but we did enjoy visits to Rhodes Old Town (if you are there around 12-30 on a sunday I believe there is a parade by the Knights of St John. Unfortunately we didn't arrive until 1pm and missed it
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) and Faliraki and one of the corner tavernas in Archangelos is a great place to have a cooling drink and watch the world go by.
We didn't find Faliraki to be the 18/30 resort we had been warned it was (look out for Subway and KFC!!) and we enjoyed our night there. Daytimes we toured and visited all manner of places. If you can get to some of the Inland villages you will find a whole different way of life - well worth the effort.
Loads of boat trips are available from various venues around the Island which usually include either a barbecue or a stop at a resort with shore side tavernas. Also, you can hire your own motor boat and spend the day travelling wherever you fancy.
I'm sure you will enjoy Rhodes and be back again
Yammas
Dave